A Property of Indefinitely Differentiable Classes

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MA THEMA TICS: W. J. TRJITZINSK7Y

719

4 W. J. Trjitzinsky, "Analytic Theory of Linear g-Difference Equations," hereafter referred to as (T). This paper will appear in the Acta Mathematica. ' For W(u), W'(l), W' more general subregions of W(u), WQ), W, respectively, can be taken (cf. (T; § 2)). 6 Concerning iterations cf. G. D. Birkhoff, "General Theory of Linear Difference Equations," Trans. Am. Math. Soc., 12, 243-284 (1911) and Birkhoff and Trjitzinsky, "Analytic Theory of Singular Difference Equations," Acta Mathematca, 60, 1-2, 1-89 (1932). The latter paper will be refefred to as (BT).

A PROPERTY OF INDEFINITELY DIFFERENTIABLE CLASSES BY W. J. TRJITZINSKY DEPARTMENT OF MATHEMATICS, NORTHWESTERN UxIvERsrrY Communicated November 15, 1932

A convenient way of defining classes of functions f(x), indefinitely differentiable on an interval (a, b) [a < b], is by letting CA denote the class of functions such that | < fvAv (V = O, 1..; a <x
I<(V)(X)

=

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720

PROC. N. A. S.

specified by the property that the members of CA are determined uniquely throughout (a, b) by the values of the derivatives, of all orders, at. n(> 2) distinct points of (a, b). In fact, suppose the theorem is not true. Then there exists a nonquasi-analytic class CA with the properties whose possibility (for nonquasi-analytic classes) has been denied in the theorem. Let the xi (i = 1, 2... n; a . xi _ b) be a set of points referred to above. There exists a function f(x), (2) If'(x) I < fVA v (v = 0,1.. .; a _ x < b) such that f(v)(xo) = 0 (v = 0, 1 . . for a fixed x0 in (a, b), while f(x) W 0. Suppose, as is possible without any loss of generality, that a = 0 and b = 1. Let p(x) be a polynomial, with real coefficients and different from a constant, such that

(i

p (Xi) =0

=

1, 2 .......

. n). .(3)

Form the function

(4)

(X)= sin2 (p(x) + b), where sin2 b = x0. Then O < s(P x)

(O :!! x < 1), .p(xi) = xo (i = 1, 2.. n). It is known that, if a(x) is analytic on (0, 1), < 1

(4a)

Iakv)(x)I< (av)v

(v = 0,1...; 0< x 1) where a is a constant depending on a(x).4 Accordingly, (.pv)' (v = 0, 1...; 0 < x < 1) I (X) and, for a suitable s°i, ,(') (4b) (x) I <_ o(Iv! (v 0, 1... ; 0 . x 1). Let M, be the upper bound of f(v) (x) for 0 < x _ 1. On noting that f v)(xo) = 0 (v = 0, 1 ...), it is concluded that

|

f~(z)

=

- - lf(m)(s)ds =( -v- 1)!,J (z s)vm -

(v

<

i).

Hence it is observed that

|f(v (Z) I -_ MD

_

(m

[) (v < m; 0 < z < 1)

(5)

(compare with (C; p. 22)). The following fact, stated in (C; p. 22), will be now noted (with modified notation). Suppose that

VOL. 18, 1932

MA THEMA TICS: W. J. TRJITZINSK Y

a _ p (x)< b If(V)(z) < B,

s(v) (x)

721

(for x _<x _ xi), (for a < z < b; v = O, ...m), (forxI _ Al < xi; v = 0, 1. . .m).

|.< C,,

Let

fp* (Z)

v.Z) + (v-

V.,

Then

dxm

dxm

(XO

m

Consider the function g(x) = f(,p(x)). In the above-mentioned device

weleta=x=0,b=xi=1, B-= (m V)!.(v=m1 * m; ef. (5)) and C, = plvv! (v = 0, 1 . .m; cf. (4b)). Thus, for f (z) and p* (x) the *

following functions may be chosen

(Z) ==O f* fm( E (mmmv! Mm)!!.

mm

M! (1 + Z)-,

=

0

V

(x)

<

Z .iX' = 1

=

J(P

V=1

Then

()]= Mm (1

f*[

-

401X)m.

Consequently, for 0 _ x <_ 1,

mm "g{~m(m + 1)... .(2m -1)

jg(m)(x)

m! { g)

=(PiMm

. (2

!

- 1) <

(1

-!pIX)2m

X-o

(2e(pi)mM. < (2evif)mAm (mf 0, 1, 2 ...). =

(2e(p1f) is independent of the choice of the set of points X2 (X1, ...Xs) Accordingly, g(x) is of the class CA. Since f(x) W 0, it follows that g(x) W 0. Moreover, in virtue of (4a) and since f(v) (xo) = 0 (v = 0, I Here the constant

we conclude that (') (xi)

=0

(i = 1, 2 ...n; v = 0, 1, 2 ...).

However, existence of such a function in the class CA means, in virtue of the additive property of CA, that the members of this class are not uniquely determined throughout the interval of definition by the values of the

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722

PROC. N. A. S.

derivatives of all orders at n(> 2) distinct points of the interval. This completes the proof of the theorem. In a different formulation this theorem can be stated as follows. For quasi-analiticity of CA (in (a,b)) it is necessary and sufficient that the functions of CA should be determined uniquely (in (a,b)) by the values of the derivatives of all orders at n(> 2) distinct points (of (a,b)). 1 This is the definition used by T. Carleman in his book Les fonctions quasi

analytique, Paris, 1926, hereafter referred to as (C). 2 Also see t. Borel, Lejons sur les fonctions monoganes, Paris, 1917, and W. J.

Trjitzinsky's papers on quasi-analytic functions in the Annals of Math., 30, 526-546 (1929); 32, 623-658 (1931); 32, 659-85 (1931). 3The derivative of order O,f of f(x) is f(x). 4Cf., for instance, C. de la Vall6e Poussin, "On the Approximation. . .," The Rice Institute Pamphlet, Vol. XII, April, 147 (1925).

A0,

OPERA TORS OF ORDER pm IN THE GROUP OF ISOMORPHISMS OF THE ABELIAN GROUP OF ORDER pn AND TYPE 1, 1, ... By H. R. BRAHANA DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS

Commuficated October 19, 1932

We propose to characterize the operators in question in terms of the well-known invariants of linear transformations. We take a group H of order pn and type 1, 1, . . . and represent one of its automorphisms as a non-homogeneous linear transformation on the exponents of a set of independent generators. It has been shown in a recent paper' that if the isomorphism U of order p transforms one of the operators of the abelian group Hk of order pk and type 1, 1, ... into a set of generators of Hk, then U may be written in the form

'O 1 0 .. O 0 1

as .I .

a,al

where aj = - (

+

...

k

O ak

j),r = p-1-k a 0.

Dickson shows'

that every linear transformation on n variables may be written in a canonical form which is a set of transformations of the type (1) on distinct chains of variables. The variables of a chain are linearly independent, and no set of variables from one chain is linearly dependent on